X-Ray Quasi-periodic Eruptions Driven by Star–Disk Collisions: Application to GSN069 and Probing the Spin of Massive Black Holes

X-ray quasi-periodic eruptions (QPEs) are discovered recently in active galaxies with unknown driven mechanism. Under the assumption that QPEs caused by star-disc collisions, we adopt full relativistic method and show both orbital parameters of star also mass spinning massive black hole (MBH) can be revealed using time arrival (TOA) QPEs. By applying model to observed GSN069, find is a near-circular orbit ( $e_\bullet=0.05^{+0.02}_{-0.02}$) semimajor axis $\sim 365^{+54}_{-49}r_{\rm g}$ around MBH $M_\bullet=3.0^{+0.9}_{-0.6} \times10^5M_\odot$. The alternative short long recurring GSN069 well explained small eccentricity precession star. We possibly striped stellar core colliding accretion disc after partial tidal disruption event MBH. For GSN069-like galaxies, if continuous monitoring QPE events accumulated uncertainties TOA $\lesssim 100-150$s, spin constrained fitting Our results timing provide unique probe for measuring tests no-hair theorem.